Decision making as a model

Decision making as a model 4. Signal detection: models and measures

Nice theorems, but how to proceed in practice? • Hard work • Get several points on ROC-curve by inducing several criteria (pay-off, signal frequency) Compute hit rate and false alarm rate at every criterion Many trials for every point! Measure or compute A using graphical methods

Certainly 0 1 2 3 4 5 Certainly no signal a signal Variant: numeric (un)certainty scale: implies multiple criteria – consumes many trials too

Hits 1-H F 1 - ¼ ------- + ----- 1-F H • (H-F)(1+H-F) • = ½ + ¼ ----------------- • H(1-F) FalseAlarms Average of those two areas: A' = • Rough approximation • Area measure for one point: A' H F Iff H>F

B''= -.4 B''= -.07 B''= 0 B''= .07 HIT RATE B''=.4 FALSE ALARM RATE if H = 1, F≠0, F≠1, then B'' = -1 F H Comparable measure for criterion/bias: Grier’s B'' Isobias curves if H = 1 - F then B'' = 0 if F = 0, H≠ 0, H≠1 then B'' = 1 H(1 - H) – F(1 – F) B'' = sign(H - F) ------------------------ H(1 - H) + F(1 – F)

Introducing assumptions • Even when several points ara available, they may not lie on a nice curve Then you might fit a curve, but which one? Every curve reflects some (implicit) assumptions about distributions Save labor: more assumptions  less measurement (but the assumptions may not be justified)

Normal distributions are popular (there are other models!) Simplest model: noise and signal distributions normal with equal variance One point (PH, PFA pair) is sufficient

Example: in an experiment with noise trials and signal trials these results were obtained: Hit rate: .933, False Alarm rate .309 (.067 misses and .691 correct rejections) Normal distributions: via corresponding z-scores the complete model can be reconstructed:

distance: d´ = 2 measure for “sensitivity” z.933 = - 1.5 f z.309 = .5 h .933 h β = ---- = .37 f Measure for bias/criterion .309

several d'-s and corresponding ROC-curves

zΦ(z) = -∞ φdx ∫ 1 φ(z)= e-z2/2√2π Gaussian models: preliminary Standaard normal curve M=0, sd = 1 P z Transformations: Φ(z)  P Φ-1(P) or: Z(P)  z see tabels and standard software

Roc-curve PH = f(PFA) PH PFA λ zH - Z-transformation ROC-curve P  z zH = f(zFA) zFA Nice way to plot several (PFA, PH) points

Plotting with regression line? Regression line zH = a1zFA+ b1, Minimize (squared) deviations zH : Underestimation of a Regresionline zFA= a2zH + b2 Minimize (squared) deviations zFA : Overestimation of a Compromise: average of regression lines ZH = ½(a1+1/a2)ZFA + ½(b1+b2/a2)

Equal variance model: PFA = 1- Φ(λ), = Φ(-λ), zFA = -λ PH = 1 – Φ(-(d' - λ)) = Φ(d'– λ), zH = d' – λ zH =zFA + d' d' = zH –zFA z-plot ROC 45° 0 λ zH d' d' 45° zFA

Criterion/bias: β = h/f = φ(zH)/φ(zF) f h 1 -z2/2 φ(z) = ------ e (standard-normal) √(2π) 1 -zH2/2 φ(zH) = ------ e √(2π) zFA2 – zH2 ------------ 2 Divide: -------------------- = e 1 -zFA2/2 φ(zFA) = ------ e √(2π) To get symmety a log transformation is often applied: log β = log h – log f = ½(z2FA – z2H )

zH – zFA 2zFA c = - ---------- + ----- 2 2 f h λ c Alternatve: c (aka λcenter), distance (in sd) between middle (were h=f) and criterion c = -(d'/2 – λ) zFA = -λ d' = zH - zFA zH + zFA c = - ---------- 2

β c Isobias curves for βen c

Decision making as a model